Abstract

We consider error estimates of iterative algorithm in shift-invariant signal spaces. For the classical sampling and reconstruction algorithm, error estimate from its samples corrupted by white noises are widely studied, but the error analysis of noise with time jitter and iterative noise has not been given as much attention. In this paper, three types of error estimates are studied. In detail, we obtain the error estimate for reconstructing a signal from its noise samples, noise samples with time jitter, and iterative noise.

1. Introduction

The famous Shannon sampling theorem [1] successful resolves the reconstruction of a function on from its samples , where is a countable index set. This is a common task in many applications in signal or image processing. The Shannon sampling theorem says that if is the bandlimited signal of finite energy, then it is completely characterized by its samples. In many engineering applications, such as MRI imaging, signals and images are not band limited. One such example is the shift-invariant spaces. Nonuniform sampling and reconstruction problems in shift-invariant spaces are a relatively recent and active research field [2–12]. The shift-invariant space model developed in the 1990s is successful for many engineering problems, where the signal to be reconstructed is assumed to live in a shift-invariant space. It has been shown to be suitable and realistic, especially for taking into account of realistic environment, for modeling signals with smooth spectrum, or for numerical implementation [8, 10, 12].

In this paper, we will assume that the functions or signals all belong to shift-invariant space of the form [2–12]

Many reconstruction algorithms are studied in shift-invariant spaces. For example, the iterative algorithm is obtained in shift-invariant spaces [3]. However, the reconstructing a function from data corrupted by noise has not been given as much attention. Smale and Zhou reconstructed signals from noisy data in [7] and gave error estimates for the reconstructed signal [8]. Aldroubi et al. discussed error analysis of frame reconstruction from noisy samples in [2]. Chen et al. gave the estimate of aliasing error for reconstruction algorithm in shift-invariant spaces [4]. We will study error analysis of the iterative reconstruction algorithm from noisy samples in shift-invariant spaces.

In this paper, the following three types of errors are considered for the iterative algorithm in shift-invariant spaces.(1)Signal samples are affected by some additive noise and are therefore given by where is sampling point and is a vector of noise samples.(2)Add time-jitter error in the above error model where are jitter. Due to the sampling time, points are not met correctly but might differ from the exact ones by not more than a given , that is, .(3)Numerical error in the th iterative step in the iterative algorithm as shown in Theorem 2.1 is , that is,

For the first kind of error analysis, it is widely studied, see [1, 3, 7, 8, 10]. However, the second and third kinds of error analysis have not been given as much attention. The second type of error analysis is presented in band-limited signal spaces [13]. The time-jitter errors without additive noise are studied in [6, 9]. Some results of the third type of error analysis are shown in reproducing kernel spaces [14]. In this paper, we study the three types of error for the iterative algorithm in shift-invariant spaces.

The paper is organized as follows. In Section 2, we introduce some concepts such as bounded partition of unity, shift-invariant spaces, and the iterative algorithm in shift-invariant spaces. In Section 3, we give error estimates of the iterative algorithm reconstruction algorithm from noisy samples.

2. Notations and Preliminaries

The shift-invariant spaces under consideration are of the form where is the so-called generator. The generator belongs to a subspace of continuous functions of Wiener amalgam spaces . A measurable function belongs to Wiener amalgam spaces if it satisfies For any , we have the norm equivalences in [3]; that is, there exist constants and such that

A set satisfying is called separated, where is a countable index set. For the sampling set , we can give the following definition of bounded partition of unity (BPU). A bounded partition of unity (BPU) associated with is a set of functions that satisfy(1), where ,(2),(3).

The operator defined by is a quasi-interpolant of the sampled values .

Aldroubi and Gröchenig presented the following iterative algorithm in [3].

Theorem 2.1. Let be sampling set with for some and be a BPU associated with . Let be in and be a bounded projection from onto . Then, there exists density such that if is separated, then any can be recovered from its samples on sampling set by the iterative algorithm

3. Error Estimates of Iterative Reconstruction Algorithm

First, for given data of the form , we give the estimation of in Theorem 3.3, where is recovered via the iterative approximation projection reconstruction algorithm in Theorem 2.1 from corrupted samples . If are noise with zero mean and variance, then the error estimates of and are presented in Theorem 3.1. Second, we discuss the second type of error. The estimate of will be shown in Theorem 3.4, where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples . Third, if there exists the numerical error in th iterative step in the iterative algorithm, then we will consider the numerical stability of the iterative algorithm in Theorem 3.5.

Theorem 3.1. Suppose that the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise . Assume that , are noises with zero mean and variance, that is, If there exists a constant such that then for any where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples and , and are defined in Theorem 2.1.

Proof. Applying (2.4) iteratively leads to which together with the convergence of (2.4) implies that where , .
Combining (3.5) with leads to
By , (3.2), and (3.6), we obtain

Remark 3.2. If we add some restricted conditions presented in [14] for generator , that is, where , and is the dual of , then it is easy to check that (3.2) holds in shift-invariant space. Song et al. obtained the restricted conditions such that (3.2) holds for bandlimited space in [15]. From Lemma  2.1 of [16], (3.2) holds for special shift-invariant, that is spline subspace.

Theorem 3.3. If the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise , then is bounded by the norm of the noise. More precisely, where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples and .

Proof. Let where .
From the iterative (3.10),
Using Lemma  8.3 of [3], we may choose so small that
By (3.4),(3.11), and (3.12), we have This implies that .

Theorem 3.4. Assume that the initial samples in the iterative approximation projection reconstruction algorithm as shown by Theorem 2.1 are corrupted, that is, for noise with and . If there exist constants such that for the generator and sampling set , then where is recovered via the iterative approximation projection reconstruction algorithm from corrupted samples as defined in Theorem 3.1 and as defined in (2.3).

Proof. From (3.2) and (3.5), we have
Next, we will estimate .
By the Hölder inequality and (3.14),
The last inequality follows from (2.3).
So, we have

Lastly, we will consider the numerical stability of the iterative algorithm.

Theorem 3.5. Assume that the numerical error in th iterative step in the iterative algorithm as shown in Theorem 2.1 is , that is, Then, we have the following estimation where .

Proof. By induction, we obtain , where and for .
Using Lemma  8.3 of [3], we may choose so small such that (3.12) holds.
Therefore,
The iterative algorithm converges exponentially too, precisely
In fact, by the iterative algorithm From (3.12), we have .
Combining (3.21) with (3.22), we have

Acknowledgments

This project is partially supported by the National Natural Science Foundation of China (10801136, 10871213), the Natural Science Foundation of Guangdong Province (07300434), and the Fundamental Research Funds for the Central Universities (10lgpy27). This work is partially done when the author is visiting the Department of Mathematics, University of Central Florida. The author thanks the department for the hospitality. The authors would like to thank Professor Qiyu Sun and Professor Wenchang Sun for their discussion and suggestions.